Translating a mixed-integer programming formulation to Scipy - python-3.x

Question

I would like to solve the above formulation in Scipy and solve it using milp(). For a given graph (V, E), f_ij and x_ij are the decision variables. f_ij is the flow from i to j (it can be continuous). x_ij is the number of vehicles from i to j. p is the price. X is the available number vehicles in a region. c is the capacity.
I have difficulty in translating the formulation to Scipy milp code. I would appreciate it if anyone could give me some pointers.
What I have done:
The code for equation (1):
f_obj = [p[i] for i in Edge]
x_obj = [0]*len(Edge)
obj = f_obj + v_obj
Integrality:
f_cont = [0 for i in Edge] # continous
x_int = [1]*len(Edge) # integer
integrality = f_cont + x_int
Equation (2):
def constraints(self):
b = []
A = []
const = [0]*len(Edge) # for f_ij
for i in v: # for x_ij
for e in Edge:
if e[0] == i:
const.append(1)
else:
const.append(0)
A.append(const)
b.append(self.accInit[i])
const = [0]*len(Edge) # for f_ij
return A, b
Equation (4):
[(0, demand[e]) for e in Edge]

Answer 1

I'm going to do some wild guessing, given how much you've left open to interpretation. Let's assume that
this is a maximisation problem, since the minimisation problem is trivial
Expression (1) is actually the maximisation objective function, though you failed to write it as such
p and d are floating-point vectors
X is an integer vector
c is a floating-point scalar
the graph edges, since you haven't described them at all, do not matter for problem setup
The variable names are not well-chosen and hide what they actually contain. I demonstrate potential replacements.
import numpy as np
from numpy.random._generator import Generator
from scipy.optimize import milp, Bounds, LinearConstraint
import scipy.sparse
from numpy.random import default_rng
rand: Generator = default_rng(seed=0)
N = 20
price = rand.uniform(low=0, high=10, size=N) # p
demand = rand.uniform(low=0, high=10, size=N) # d
availability = rand.integers(low=0, high=10, size=N) # X aka. accInit
capacity = rand.uniform(low=0, high=10) # c
c = np.zeros(2*N) # f and x
c[:N] = -price # (1) f maximized with coefficients of 'p'
# x not optimized
CONTINUOUS = 0
INTEGER = 1
integrality = np.empty_like(c, dtype=int)
integrality[:N] = CONTINUOUS # f
integrality[N:] = INTEGER # x
upper = np.empty_like(c)
upper[:N] = demand # (4) f
upper[N:] = availability # (2) x
eye_N = scipy.sparse.eye(N)
A = scipy.sparse.hstack((-eye_N, capacity*eye_N)) # (3) 0 <= -f + cx
result = milp(
c=c, integrality=integrality,
bounds=Bounds(lb=np.zeros_like(c), ub=upper),
constraints=LinearConstraint(lb=np.zeros(N), A=A),
)
print(result.message)
flow = result.x[:N]
vehicles = result.x[N:].astype(int)